Generally, in optics caustics are patterns of light created by reflection or refraction on curved surfaces. Computing caustics in a given three dimensional scene has been the subject of extensive research in computer graphics as it constitutes one of the main difficulties of photorealistic rendering algorithms.
The task of reproducing a pre-specified light distribution by a specular surface also arises in the field of inverse reflector design, which concentrates on reflectors for lamps. A survey on inverse reflector design is given by Patow and Pueyo [PP05; Patow et al; 2005]. Generally, such light distributions can be classified as either near-field or far-field distributions.
While near-field distributions specify an irradiance distribution on a given surface (typically a plane) that is to be reproduced, far-field distributions can be considered as limit cases where the surface to be illuminated is infinitely far away from the reflector, so that only the distribution of the ray directions matters. Methods for inverse reflector design typically employ an analysis-by-synthesis approach. A certain surface representation is chosen to parametrize the reflector, such as NURBS [ASG08; Anson et al. 2008]. Then, the light distribution caused by a surface is evaluated and rated against the desired one. This method is iteratively used to optimize the surface parameters. Various optimization strategies have been applied, including frameworks that allow an analytical differentiation, thereby enabling the use of the conjugate gradient method [Neu97; Neubauer 1997], and methods that compute derivatives approximately [FDL10; Finckh et al. 2010] to ones that employ no derivatives at all [ASG08; Anson et al. 2008].
Examples using an evolutionary optimization [DCC99; Doyle et al. 1999] also belong to the latter category. Common simplifications in the approaches are the assumption of perfect specularity of the surface and the assumption of only one bounce of light without interreflections or occlusions, although exceptions to both also exist [PPV07; Patow et al. 2007; MMP09; Mas et al. 2009]. The restriction to rotationally symmetric reflectors is also commonly used, particularly in theoretical works [WN75; Westcott et al. 1975].
These works mostly focus on reflective surfaces, although many approaches readily can be extended to refraction as well. One noteworthy example investigating the problem of refraction is the work by [FDL10; Finckh et al. 2010]. They use GPU computations to speed up the caustic evaluation, and a stochastic approximation algorithm for the optimization, which is able to find a global optimum.
Concerning refractive objects, the field of lens design is also noteworthy, although the goals of these problems are different, e.g. aberration correction. These problems are often restricted to a small number of parameters such as radii of the underlying primitive shapes [PP05; Patow et al. 2005]. Again, there are exceptions, e.g. the work by [LSS98; Loos et al. 1998], who use a NURBS-based representation to optimize progressive lenses.
[WPMR09; Weyrich et al. 2009] have chosen a different approach to reproduce a pre-specified far-field distribution. First, they generated a set of sloped, planar microfacets to realize the desired distribution of ray directions. Then, they arranged the microfacets in a regular array using simulated annealing to minimize the resulting discontinuities. Closely related to Weyrich et al.'s work is the system for near-fields proposed by [PJJ+11; Papas et al. 2011]. They extended the notion of microfacets to curved micropatches, which are used to produce specks of light with an anisotropic Gaussian distribution. To compute the shape of the micropatches that produce a Gaussian irradiance distribution, Papas et al. define a bijective mapping between points in the micropatch domain and points on the projection plane, analytically compute the surface normals that refract/reflect the light in this way, and finally integrate this normal field to arrive at the required micropatch surface.
It is an object of the present invention to provide a method for producing a reflective or refractive surface that reflects or refracts light shined thereon and reproduces on a screen a desired greyscale intensity image on which the reflective or refractive surface is based and a corresponding apparatus, wherein the method permits a reproduction of a reference grayscale image with adjustable precision.